We introduce and analysis the `target map' for the shooting method.
For a large class of elliptic systems as well as more general dynamic
systems, we show that the target map is onto via the degree theo-
ry. The target map is onto implies that we can shoot to any desired
position.
Applying our result to a motivating example, we obtain the ex-
istence of global positive solutions to the Hardy-Littlewood-Sobolev
(also known as Lane-Emden) type system.

in the critical and supercritical case. On the other
hand, the Lane-Emden conjecture state that when k = 1, the above
system admits no positive solution in the subcritical case.

The Lane-emden conjecture has been proved for many cases
and no counter-example has been found yet.
This new approach to the shooting method is simple and powerful.
We expect many more applications of it.